#489928

order	512
nilpotency class	4
isoclinism	isoclinic class of #489745
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

641 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 80 points:

(1,2)(3,12)(4,15)(5,11)(6,10)(7,9)(8,13)(14,16)(17,18,23,28)(19,29,32,59)(20,25,34,49)(21,37,42,57)(22,43,40,27)(24,45,47,31)(26,52,55,41)(30,60,61,58)(33,65,44,53)(35,67,39,72)(36,70,66,51)(38,56,74,48)(46,64,76,62)(50,75,54,69)(63,80,71,78)(68,79,73,77);(1,3,7,4)(2,8,9,16)(5,14,6,13)(10,15,11,12)(17,19,23,32)(18,24,28,47)(20,30,34,61)(21,38,42,74)(22,44,40,33)(25,46,49,76)(26,53,55,65)(27,56,43,48)(29,57,59,37)(31,52,45,41)(35,68,39,73)(36,71,66,63)(50,78,54,80)(51,79,70,77)(58,67,60,72)(62,75,64,69);(1,4,7,3)(2,8,9,16)(5,13,6,14)(10,15,11,12)(17,20,23,34)(18,25,28,49)(19,30,32,61)(21,39,42,35)(22,36,40,66)(24,46,47,76)(26,54,55,50)(27,51,43,70)(29,58,59,60)(31,64,45,62)(33,63,44,71)(37,72,57,67)(38,73,74,68)(41,75,52,69)(48,77,56,79)(53,80,65,78);(2,9)(8,16)(10,11)(12,15)(20,34)(25,49)(30,61)(35,39)(36,66)(46,76)(50,54)(51,70)(58,60)(62,64)(63,71)(67,72)(68,73)(69,75)(77,79)(78,80)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397124);